Convergence to the equilibria for self-stabilizing processes in double-well landscape

Introduction

We investigate the weak convergence in long-time of the following so-called self-stabilizing process:

Here, $\ast$ denotes the convolution. Since the own law of the process intervenes in the drift, this equation is nonlinear, in the sense of McKean. We note that $X_{t}$ depends on $\varepsilon$. We do not write $\varepsilon$ for simplifying the reading.

This kind of processes were introduced by McKean, see McKean or McKean1966 . Here, we will make some smoothness assumptions on the interaction potential $F$. Let just note that it is possible to consider nonsmooth $F$. If $F$ is the Heaviside step function and $V:=0$, (I) is the Burgers equation; see SV1979 . If $F:=\delta_{0}$, and without confining potential, it is the Oelschläger equation, studied in Oel1985 .

The particle $X_{t}$ which verifies (I) can be seen as one particle in a continuous mean-field system of an infinite number of particles. The mean-field system that we will consider is a random dynamical system like

The link between the self-stabilizing process and the mean-field system when $N$ goes to $+\infty$ is called the propagation of chaos; see Sznitman under Lipschitz properties; BRTV if $V$ is a constant; Malrieu2001 or Malrieu2003 when both potentials are convex; BAZ1999 for a more precise result; BGV2007 , DPPH1996 or DG1987 for a sharp estimate; CGM for a uniform result in time in the nonuniformly convex case.

Equation (II) can be rewritten in the following way:

where the $i$th coordinate of $\mathcal{X}_{t}$ (resp., $\mathcal{B}_{t}$) is $X_{t}^{i}$ (resp., $B_{t}^{i}$) and

As observed in DG1987 , the empirical law of the mean-field system can be seen as a perturbation of the law of the diffusion (I). Consequently, the long-time behavior of $\mathcal{L}(X_{t})$ that we study in this paper provides some consequences on the exit time for the particle system (II).

Also, the convergence plays an important role in the exit problem for the self-stabilizing process since the exit time is strongly linked to the drift according to the Kramers law (see DZ or HIP ) which converges toward a homogeneous function if the law of the process converges toward a stationary measure.

Let us recall briefly some of the previous results on diffusions like (I). The existence problem has been investigated by two different methods. The first one consists in the application of a fixed point theorem; see McKean , BRTV , CGM or HIP in the nonconvex case. The other consists of a propagation of chaos; see, for example, M1996 . Moreover, it has been proved in Theorem 2.13 in HIP that there is a unique strong solution.

In McKean , the author proved—by using Weyl’s lemma—that the law of the unique strong solution $du_{t}^{\varepsilon}$ admits a $\mathcal{C}^{\infty}$-continuous density $u_{t}^{\varepsilon}$ with respect to the Lebesgue measure for all $t>0$. Furthermore, this density satisfies a nonlinear partial differential equation of the following type:

It is then possible to study equations like (III) by probabilistic methods which involve diffusions (I) or (II); see CGM , Funaki1984 , Malrieu2003 . Reciprocally, equation (III) is a useful tool for characterizing the stationary measure(s) and the long-time behavior; see BRTV , BRV , Tamura194 , Tamura1987 , Veret2006 . In HT1 , in the nonconvex case, by using (III), it has been proved that the diffusion (I) admits at least three stationary measures under assumptions easy to verify. One is symmetric, and the two others are not. Moreover, Theorem 3.2 in HT1 states the thirdness of the stationary measures if $V^{\prime\prime}$ is convex and $F^{\prime}$ is linear. This nonuniqueness prevents the long-time behavior from being as intuitive as in the case of unique stationary measure.

The work in HT2 and HT3 provides some estimates of the small-noise asymptotic of these three stationary measures. In particular, the convergence toward Dirac measures and its rate of convergence have been investigated. This will be one of the two main tools for obtaining the convergence.

Convergence for (I) is not a new subject. In BRV , if $V$ is identically equal to , the authors proved the convergence toward the stationary measure by using an ultracontractivity property, a Poincaré inequality and a comparison lemma for stochastic processes. The ultracontractivity property still holds if $V$ is not convex by using the results in KKR . It is possible to conserve the Poincaré inequality by using the theorem of Muckenhoupt (see logsob2000 ) instead of the Bakry–Emery theorem. But, the comparison lemma needs some convexity properties. However, it is possible to apply these results if the initial law is symmetric in the synchronized case ($V^{\prime\prime}(0)+F^{\prime\prime}(0)\geq 0$); see Theorem 7.10 in TT .

Another method consists of using the propagation of chaos in order to derive the convergence of the self-stabilizing process from the one of the mean-field system. However, we shall use it independently of the time and the classical result which is on a finite interval of time is not sufficiently strong. Cattiaux, Guillin and Malrieu proceeded a uniform propagation of chaos in CGM and obtained the convergence in the convex case, including the nonuniformly strictly convex case. See also Malrieu2003 . Nevertheless, according to Proposition 5.17 and Remark 5.18 in TT , it is impossible to find a general result of uniform propagation of chaos. In the synchronized case, if the initial law is symmetric, it is possible to find such a uniform propagation of chaos; see Theorems 7.11 and 7.12 in TT .

The method that we will use in this paper is based on the one of BCCP . See also Malrieu2003 , Tamura194 , Malrieu2001 , HS1987 , AMTU2001 for the convex case. In the nonconvex case, Carrillo, McCann and Villani provide the convergence in CMV2003 under two restrictions: the center of mass is fixed and $V^{\prime\prime}(0)+F^{\prime\prime}(0)>0$ (that means it is the synchronized case).

However, by combining the results in HT1 , HT2 , HT3 with the work of BCCP (and the more rigorous proofs in CMV2003 about the free-energy), we will be able to prove the convergence in a more general setting. The principal tool of the paper is the monotonicity of the free-energy along the trajectories of (III).

First, we introduce the following functional:

This quantity appears intuitively as the limit of the potential in (II) for $N\to+\infty$. We consider now the free-energy of the self-stabilizing process (I),

for all measures $du$ which are absolutely continuous with respect to the Lebesgue measure. We can note that $du_{t}^{\varepsilon}$ satisfies this hypothesis for all $t>0$.

The paper is organized as follows. After presenting the assumptions, we will state the first results, in particular, the convergence of a subsequence $(u_{t_{k}}^{\varepsilon})_{k}$. This subconvergence will be used for improving the results about the thirdness of the stationary measures. Then, we will give the main statement which is the convergence toward a stationary measure, briefly discuss the assumptions of the theorem and give the proof. Subsequently, we will study the basins of attraction by two different methods and prove that these basins are not reduced to a single point. Finally, we postpone four results in the annex, including Proposition .2 which extends the classical higher-bound for the moments of the self-stabilizing processes.

We assume the following properties on the confining potential $V$ (see Figure 1): {longlist}[(V-1)]

$V$ is an even polynomial function with $\deg(V)=:2m\geq 4$.

The equation $V^{\prime}(x)=0$ admits exactly three solutions: $a$, $-a$ and with $a>0$. Furthermore, $V^{\prime\prime}(a)>0$ and $V^{\prime\prime}(0)<0$. Then, the bottoms of the wells are located in $x=a$ and $x=-a$.

$\lim_{x\to\pm\infty}V^{\prime\prime}(x)=+\infty$ and $V^{\prime\prime}(x)>0$ for all $x\geq a$.

Let us remark that the positivity of $V^{\prime\prime}$ on $[-a;a]^{c}$ [in hypothesis (V-4)] is an immediate consequence of (V-1) and (V-5). The simplest and most studied example is $V(x):=\frac{x^{4}}{4}-\frac{x^{2}}{2}$. Also, we would like to stress that weaker assumptions could be considered, but all the mathematical difficulties are present in the polynomial case, and it allows us to avoid some technical and tedious computations. Let us present now the assumptions on the interaction potential $F$: {longlist}[(F-1)]

$F$ is an even polynomial function with $\deg(F)=:2n\geq 2$.

Initialization: $F(0)=0$. Under these assumptions, we know by HT1 that (I) admits at least one symmetric stationary measure. And, if $\sum_{p=0}^{2n-2}\frac{|F^{(p+2)}(a)|}{p!}a^{p}<F^{\prime\prime}(0)+V^{\prime\prime}(a)$, there are at least two asymmetric stationary measures: $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$. Furthermore, we know by HT2 that there is a unique nonnegative real $x_{0}$ such that $V^{\prime}(x_{0})+\frac{1}{2}F^{\prime}(2x_{0})=0$ and $V^{\prime\prime}(x_{0})+\frac{F^{\prime\prime}(0)+F^{\prime\prime}(2x_{0})}{2}>0$. The same paper provides that $u^{\varepsilon}_{0}$ converges weakly toward $\frac{1}{2}\delta_{x_{0}}+\frac{1}{2}\delta_{-x_{0}}$ and $u^{\varepsilon}_{\pm}$ converges weakly toward $\delta_{\pm a}$ in the small-noise limit.

We present now the assumptions on the initial law $du_{0}$: {longlist}[(ES)]

The $8q^{2}$th moment of the measure $du_{0}$ is finite with $q:=\max\{m,n\}$.

$V^{\prime\prime}(0)+F^{\prime\prime}(0)>0$.

The process (I) admits exactly three stationary measures. One is symmetric: $u^{\varepsilon}_{0}$ and the other ones are asymmetric: $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$. Furthermore, $\Upsilon_{\varepsilon}(u^{\varepsilon}_{+})=\Upsilon_{\varepsilon}(u^{\varepsilon}_{-})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$.

There exists $M>0$ such that the diffusion (I) admits exactly three stationary measures with free-energy less than $M$. Furthermore, we have $\Upsilon_{\varepsilon}(u^{\varepsilon}_{+})=\Upsilon_{\varepsilon}(u^{\varepsilon}_{-})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})\leq M$ ; $u^{\varepsilon}_{0}$ is symmetric, and $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$ are asymmetric.

The process (I) admits only one symmetric stationary measure $u^{\varepsilon}_{0}$. In the following, we will give some simple conditions such that (M3), (M3)′ or (0M1) are true.

Finally, we recall assumption (H) introduced in HT2 : {longlist}

For concluding the Introduction, we write the statement of the main theorem:

Let $du_{0}$ be a probability measure which verifies (FE) and (FM). Under (M3), $u_{t}^{\varepsilon}$ converges weakly toward a stationary measure.

First results

This section is devoted to present the tools that we will use for proving the main result of the paper. Furthermore, we provide some new results about the thirdness of the stationary measures for the self-stabilizing processes.

In the following, we will need two particular functions [the free-energy of the system and a function $\eta_{t}$ such that $\frac{d}{dt}u_{t}^{\varepsilon}(x)=\frac{d}{dx}\eta_{t}(x)$].

According to (III), we remark that if $\eta_{t}$ is identically equal to , then $u_{t}^{\varepsilon}$ is a stationary measure for (I).

We recall the following well-known entropy dissipation:

Let $du_{0}$ be a probability measure which verifies (FE) and (ES). Then, for all $t,s\geq 0$, we have

Furthermore, $\xi$ is derivable, and we have

We can remark that $u_{t}^{\varepsilon}\in\mathcal{M}_{8q^{2}}$ for all $t>0$; see McKean . The first tool is the Proposition 1.2 [i.e., to say the fact that the free-energy is decreasing along the orbits of (III)]. The second one is its lower-bound.

Let us recall $\Upsilon_{\varepsilon}(u)\geq\Upsilon_{\varepsilon}^{-}(u)$. It suffices then to prove the inequality $\inf_{u\in\mathcal{M}_{8q^{2}}}\Upsilon_{\varepsilon}^{-}(u)\geq\Xi_{\varepsilon}$. We proceed as in the first part of the proof of Theorem 2.1 in BCCP . We show that we can minorate the negative part of the entropy by a function of the second moment. Then a growth condition of $V$ will provide the result.

We split the negative part of the entropy into two integrals,

By definition of $I_{+}$, we have the following estimate:

By putting $\gamma(x):=\sqrt{x}\log(x)\mathbh{1}_{\{x<1\}}$, a simple computation provides $\gamma(x)\geq-2e^{-1}$ for all $x<1$. We deduce

By hypothesis, there exist $C_{2},C_{4}>0$ such that $V(x)\geq C_{4}x^{4}-C_{2}x^{2}$ so the function $x\mapsto V(x)-\frac{\varepsilon}{4}x^{2}$ is lower-bounded by a negative constant. This achieves the proof. Let us note that the unique assumption we used is $\lim_{x\to\pm\infty}V^{\prime\prime}(x)=+\infty$.

The assumption (FE) implies $\xi(0)=\Upsilon_{\varepsilon}(u_{0})<\infty$. As $\xi$ is nonincreasing by Lemma 1.2 and lower-bounded by a constant $\Xi_{\varepsilon}$ according to Lemma 1.3, we deduce that the function $\xi$ converges toward a real $L_{0}$.

If and only if $\xi^{\prime}(t)=0$, the following is true: $u_{t}^{\varepsilon}$ is a stationary measure $u^{\varepsilon}$.

If $u_{t}^{\varepsilon}$ is a stationary measure $u^{\varepsilon}$, then $\xi(t)=\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})=\Upsilon_{\varepsilon}(u^{\varepsilon})$ is a constant. This provides $\xi^{\prime}(t)=0$.

Reciprocally, if $\xi^{\prime}(t)=0$, Proposition 1.2 implies

2 Subconvergence

Let $du_{0}$ be a probability measure which satisfies the assumptions (FE) and (ES). Then there exists a stationary measure $u^{\varepsilon}$ and a sequence $(t_{k})_{k}$ which converges toward infinity such that $u_{t_{k}}^{\varepsilon}$ converges weakly toward $u^{\varepsilon}$.

Plan: First, we use the convergence of $\int_{t}^{\infty}\xi^{\prime}(s)\,ds$ toward when $t$ goes to infinity, and we deduce the existence of a sequence $(t_{k})_{k}$ such that $\xi^{\prime}(t_{k})$ tends toward when $k$ goes to infinity. Then, we extract a subsequence of $(t_{k})_{k}$ for obtaining an adherence value. By using a test function, we prove that this adherence value is a stationary measure. {longlist}[Step 1.]

The uniform boundedness of the first $8q^{2}$ moments with respect to the time allows us to use Prohorov’s theorem: we can extract a subsequence [we continue to write it $(t_{k})_{k}$ for simplifying] such that $u_{t_{k}}^{\varepsilon}$ converges weakly toward a probability measure $u^{\varepsilon}$.

This means that $u^{\varepsilon}$ is a weak solution of the equation

Now, we consider a smooth function $\widetilde{\varphi}$ with compact support $[a,b]$. We put

$\varphi$ is also a smooth function with compact support. Indeed, the application $x\mapsto F\ast u^{\varepsilon}(x)$ is a polynomial function parametrized by the moments of $u^{\varepsilon}$, and these moments are bounded. Equality (1.2) becomes

With the assumptions and the notation of Theorem 1.6, we have the following limit:

where $t\mapsto\eta_{t}$ is defined in Definition 1.1. By using (V) and the growth property of $V^{\prime}$ and $F^{\prime}$, it yields

where $C_{2}$ is a constant. By using the Cauchy–Schwarz inequality, like in the proof of Theorem 1.6, we obtain

The quantity $\sqrt{-\xi^{\prime}(t_{k})}$ tends toward , so it is bounded. Finally, it leads to

The second term is bounded as in the proof of Lemma 1.3:

Consequently, $\Upsilon_{\varepsilon}(u_{t_{k}}^{\varepsilon})$ converges toward $\Upsilon_{\varepsilon}(u^{\varepsilon})$, then $\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})$ converges toward $\Upsilon_{\varepsilon}(u^{\varepsilon})$ since the free-energy is monotonous.

By taking $R$ big enough and then $k$ big enough, we can make the following quantity arbitrarily small: $|\int u_{t_{k}}^{\varepsilon}\log(u_{t_{k}}^{\varepsilon})-\int u^{\varepsilon}\log(u^{\varepsilon})|$.

3 Consequences

When $V$ is symmetric, Proposition 3.1 (resp., Theorem 4.6) in HT1 states the existence of at least three stationary measures for $\varepsilon$ small enough if $F^{\prime}$ is linear [resp., if $\sum_{p=0}^{\infty}\frac{|F^{(p+2)}(a)|}{p!}a^{p}<F^{\prime\prime}(0)+V^{\prime\prime}(a)$]. Theorem 1.6 permits to extend these results.

For $\varepsilon$ small enough, process (I) admits at least three stationary measures: one is symmetric ($u^{\varepsilon}_{0}$), and two are asymmetric ($u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$). Moreover, for sufficiently small $\varepsilon$, $\Upsilon_{\varepsilon}(u^{\varepsilon}_{+})=\Upsilon_{\varepsilon}(u^{\varepsilon}_{-})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$.

We know by Theorem 4.5 in HT1 that there exists a symmetric stationary measure $u^{\varepsilon}_{0}$. Theorem 5.4 in HT2 implies the weak convergence of $u^{\varepsilon}_{0}$ toward $\frac{1}{2}(\delta_{x_{0}}+\delta_{-x_{0}})$ in the small-noise limit where $x_{0}\in[0;a[$ is the unique solution of

We note that $V(x_{0})+\frac{1}{4}F(2x_{0})>V(a)$. Consequently, for $\varepsilon$ small enough, we have $\Upsilon_{\varepsilon}(v^{\varepsilon}_{+})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$.

We consider now process (I) starting by $u_{0}:=v^{\varepsilon}_{+}$. This is possible because the $8q^{2}$th moment of $v^{\varepsilon}_{+}$ is finite. Theorem 1.6 implies the existence of a sequence $(t_{k})_{k}$ which goes to infinity such that $u_{t_{k}}^{\varepsilon}$ converges weakly toward a stationary measure $u^{\varepsilon}$ satisfying $\Upsilon_{\varepsilon}(u^{\varepsilon})\leq\Upsilon_{\varepsilon}(u_{0})=\Upsilon_{\varepsilon}(v^{\varepsilon}_{+})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$. So $u^{\varepsilon}\neq u^{\varepsilon}_{0}$. We immediately deduce the existence of at least two stationary measures.

If $V^{\prime\prime}(0)+F^{\prime\prime}(0)\neq 0$, we know by Theorem 1.6 in HT3 that there exists a unique symmetric stationary measure for $\varepsilon$ small enough. Hence $u^{\varepsilon}$ is not symmetric.

Let us assume now that $V^{\prime\prime}(0)+F^{\prime\prime}(0)=0$. By (1.1), and by the definition of $\Upsilon_{\varepsilon}^{-}(u)$, we have

for all $u\in\mathcal{M}_{8q^{2}}$. Since $F^{\prime\prime}$ is convex, $x\mapsto F(x)-\frac{F^{\prime\prime}(0)}{2}x^{2}$ is nonnegative. It yields

In particular, this holds for the symmetric measures. Then, for $\varepsilon$ small enough, $\Upsilon_{\varepsilon}(u)>\frac{V(a)}{2}$ for all the symmetric measures. However, $\Upsilon_{\varepsilon}(v^{\varepsilon}_{+})<\frac{V(a)}{2}$ [then $\Upsilon_{\varepsilon}(u^{\varepsilon})<\frac{V(a)}{2}$] for $\varepsilon$ small enough.

Consequently, the process admits at least one asymmetric stationary measure that we call $u^{\varepsilon}_{+}$. The measure $u^{\varepsilon}_{-}(x):=u^{\varepsilon}_{+}(-x)$ is invariant too. By construction of $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$, $\Upsilon_{\varepsilon}(u^{\varepsilon}_{+})=\Upsilon_{\varepsilon}^{-}(u^{\varepsilon}_{-})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$.

By a similar method, we could also prove the existence of at least one stationary measure in the asymmetric-landscape case.

We know by Theorem 3.2 in HT1 that if $V^{\prime\prime}$ is convex and if $F^{\prime}$ is linear, there are exactly three stationary measures for $\varepsilon$ small enough. We present a more general setting. In view of the convergence, we will prove that the number of relevant stationary measures is exactly three even if it is a priori possible to imagine the existence of at least four such measures.

We assume $F^{\prime\prime}(0)+V^{\prime\prime}(0)>0$. Then, for all $M>0$, there exists $\varepsilon(M)>0$ such that for all $\varepsilon\leq\varepsilon(M)$, the number of measures $u$ satisfying the two following conditions is exactly three: {longlist}[(1)]

$u$ is a stationary measure for the diffusion (I).

$\Upsilon_{\varepsilon}(u)\leq M$. Moreover, if $\deg(V)=2m>2n=\deg(F)$, diffusion (I) admits exactly three stationary measures for $\varepsilon$ small enough.

Plan. We will begin to prove the second statement (when $m>n$). For doing this, we will use Corollary 1.9 and the results in HT2 , HT3 . Then, we will prove the first statement by using the second one and a minoration of the free-energy for a sequence of stationary measures which does not verify (H).

Corollary 1.9 implies the existence of $\varepsilon_{0}>0$ such that process (I) admits at least three stationary measures (one is symmetric, and two are asymmetric) if $\varepsilon<\varepsilon_{0}$: $u^{\varepsilon}_{+}$, $u^{\varepsilon}_{-}$ and $u^{\varepsilon}_{0}$.

Proposition 3.1 in HT2 implies that each family of stationary measures for the self-stabilizing process (I) verifies condition (H). It has also been shown that under (H), we can extract a subsequence which converges weakly from any family of stationary measures $(u^{\varepsilon})_{\varepsilon>0}$ of the diffusion (I).

Since $F^{\prime\prime}(0)+V^{\prime\prime}(0)>0$, there are three possible limiting values: $\delta_{0}$, $\delta_{a}$ and $\delta_{-a}$ according to Proposition 3.7 and Remark 3.8 in HT2 .

As $F^{\prime\prime}(0)+V^{\prime\prime}(0)>0$ and $V^{\prime\prime}$ and $F^{\prime\prime}$ are convex, there is a unique symmetric stationary measure for $\varepsilon$ small enough by Theorem 1.6 in HT3 . Also, Theorem 1.6 in HT3 implies there are exactly two asymmetric stationary measures for $\varepsilon$ small enough. This achieves the proof of the statement.

Now, we will prove the first statement. First, if $m>n$, by applying the second statement, the result is obvious. We assume now $m\leq n$. Let $M>0$. All the previous results still hold if we restrict the study to the families of stationary measures which verify condition (H). Consequently, it is sufficient to show the following results in order to achieve the proof of the theorem: {longlist}[(1)]

$\sup\{\Upsilon_{\varepsilon}(u^{\varepsilon}_{0});\Upsilon_{\varepsilon}(u^{\varepsilon}_{+});\Upsilon_{\varepsilon}(u^{\varepsilon}_{-})\}<M$ for $\varepsilon$ small enough.

Lemma .3 tells us that $\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$ [resp., $\Upsilon_{\varepsilon}(u^{\varepsilon}_{+})=\Upsilon_{\varepsilon}(u^{\varepsilon}_{-})$] tends toward [resp., $V(a)<0$] when $\varepsilon$ goes to . Hence, the first point is obvious.

We will prove the second point. We recall lower-bound (1.1),

As $V(x)\geq C_{4}x^{4}-C_{2}x^{2}$ and $\Upsilon_{\varepsilon}^{-}(u)\leq\Upsilon_{\varepsilon}(u)$ for all smooth $u$, we obtain

By taking the notation of HT2 , we have the equality $u^{\varepsilon}(x)=Z^{-1}\exp[-\frac{2}{\varepsilon}(W_{\varepsilon}(x))]$ with

We introduce $\omega(\varepsilon):=\sup\{|\omega_{k}(\varepsilon)|^{{1}/{(2n-k)}};1\leq k\leq 2n\}$.

We note that $\omega_{2n}(\varepsilon)=\frac{V^{(2n)}(0)+F^{(2n)}(0)}{(2n)!}>0$. Then, $\omega(\varepsilon)$ is uniformly lower-bounded. Consequently, we can divide by $\omega(\varepsilon)$.

The change of variable $x:=\omega(\varepsilon)y$ provides

The $2n$ sequences $(\frac{\omega_{k}(\varepsilon)}{\omega(\varepsilon)^{2n-k}})_{\varepsilon}$ are bounded so we can extract a subsequence of $\varepsilon$ (we continue to write $\varepsilon$ for simplifying) such that $\frac{\omega_{k}(\varepsilon)}{\omega(\varepsilon)^{2n-k}}$ converges toward $\widehat{\omega}_{k}$ when $\varepsilon\to 0$. We put $\widehat{W}(x):=\sum_{k=1}^{2n}\widehat{\omega}_{k}x^{k}$. We call $A_{1},\ldots,A_{r}$ the $r\geq 1$ location(s) of the global minimum of $\widehat{W}$.

If $(\omega(\varepsilon))_{\varepsilon}$ is bounded, since the quantity $\sum_{j=1}^{r}p_{j}A_{j}^{2n}$ is finite, we deduce that $(m_{2n}(\varepsilon))_{\varepsilon}$ is bounded too. Since $m_{2n}(\varepsilon)$ tends toward infinity when $\varepsilon$ goes to , we deduce that $(\omega(\varepsilon))_{\varepsilon}$ converges toward infinity. As $m_{2}(\varepsilon)$ is bounded, the quantity $\frac{m_{2}(\varepsilon)}{\omega(\varepsilon)^{2}}$ vanishes when $\varepsilon$ goes to . This means $\sum_{j=1}^{r}p_{j}A_{j}^{2}=0$ which implies $A_{j}=0$ for all $1\leq j\leq r$. Then $\sum_{j=1}^{r}p_{j}A_{j}^{2n}=0$. Consequently, $m_{2n}(\varepsilon)=o\{\omega(\varepsilon)^{2n}\}$. The Jensen inequality provides $m_{k}(\varepsilon)=o\{\omega(\varepsilon)^{k}\}$.

We recall the definition of $\omega_{k}(\varepsilon)$,

We deduce $\omega_{k}(\varepsilon)=O\{m_{2n-k}(\varepsilon)\}=o\{\omega(\varepsilon)^{2n-k}\}$. So

The assumption (LIN) implies (M3) (and (M3)′ because it is weaker) and (0M1) for $\varepsilon$ small enough. The condition (SYN) implies (M3)′ and (0M1) for $\varepsilon$ small enough. Furthermore, if $\deg(V)>\deg(F)$, (SYN) implies (M3) when $\varepsilon$ is less than a threshold.

This description of the stationary measures permits us to obtain the principal result, that is to say, the long-time convergence of the process.

Global convergence

Let $du_{0}$ be a probability measure which verifies (FE) and (FM). Under (M3), $u_{t}^{\varepsilon}$ converges weakly toward a stationary measure.

The proof is postponed in Section 2.3. First, we will discuss briefly the assumptions.

2 Remarks on the assumptions

Consequently, it is sufficient to apply Theorem 2.1 to the probability measure $u_{1}^{\varepsilon}$ since there is a unique solution to the nonlinear equation (I).

If $V$ was convex, a little adaptation of the theorem in OV2001 (taking into account the fact that the drift is not homogeneous here) would provide the nonoptimal following inequality:

for all $t>0$. The second moment of $u_{t}^{\varepsilon}$ is upper-bounded uniformly with respect to $t$. By using the convexity of $V$ and $F$, we can prove the same thing for $v_{t}^{\varepsilon}$. Consequently, since $t>0$, the free-energy is finite so the entropy is finite. However, in this paper, we deal with nonconvex landscape, so we will not relax this hypothesis.

All the moments are finite

We make the integration with an “almost-polynomial” function because we need the square of the derivative of such function to be uniformly bounded with respect to the time.

Hypothesis (M3)

As written before, the key for proving the uniqueness of the adherence value is to proceed a reductio ad absurdum and then to construct a stationary measure $u^{\varepsilon}$ such that $\Phi(u^{\varepsilon})$ takes a forbidden value [a value different from $\Phi(u^{\varepsilon}_{0})$, $\Phi(u^{\varepsilon}_{+})$ and $\Phi(u^{\varepsilon}_{-})$].

But, it is possible to deal with a weaker hypothesis. Indeed, by considering an initial law with finite free-energy and since the free-energy is decreasing, it is impossible for $u_{t}^{\varepsilon}$ to converge toward a stationary measure with a higher energy. Consequently, we can consider (M3)′ instead of (M3).

All of these remarks allow us to obtain the following result:

Let $du_{0}$ be a probability measure with finite entropy. If $V$ and $F$ are polynomial functions such that $F^{\prime\prime}(0)+V^{\prime\prime}(0)>0$, $u_{t}^{\varepsilon}$ converges weakly toward a stationary measure for $\varepsilon$ small enough.

3 Proof of the theorem

In order to obtain the statement of Theorem 2.1, we will provide two lemmas and one proposition about the free-energy. The lemmas state that a probability measure which verifies simple properties and with a level of energy is necessary a stationary measure for the self-stabilizing process (I). The third one allows us to confine all the adherence values under a level of energy.

Under (M3), if $u$ is a probability measure which satisfies (FE) and (ES), the inequality $\Upsilon_{\varepsilon}(u)\leq\Upsilon_{\varepsilon}(u^{\varepsilon}_{\pm})$ implies $u\in\{u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}$.

Let $u$ be such a measure. We consider the process (I) starting by the initial law $u_{0}:=u$. Theorem 1.6 implies that there exists a stationary measure $u^{\varepsilon}$ such that $\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})$ converges toward $\Upsilon_{\varepsilon}(u^{\varepsilon})$.

However, according to Propositions 1.2 and 1.8,

Condition (M3) provides $u^{\varepsilon}\in\{u^{\varepsilon}_{+};u^{\varepsilon}_{-};u^{\varepsilon}_{0}\}$. But, $\Upsilon_{\varepsilon}(u^{\varepsilon})\leq\Upsilon_{\varepsilon}(u^{\varepsilon}_{\pm})<\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$ so $u^{\varepsilon}\in\{u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}$. Without loss of generality, we will assume $u^{\varepsilon}=u^{\varepsilon}_{+}$.

Consequently, the function $\xi$ (see Definition 1.1) is constant. We deduce that $\xi^{\prime}(t)=0$ for all $t\geq 0$. Lemma 1.5 implies that $u_{t}^{\varepsilon}$ is a stationary measure. This means that $u=u_{0}=u^{\varepsilon}=u^{\varepsilon}_{+}$. We have a similar result with the symmetric measures:

Under (0M1), if $u$ is a symmetric probability measure satisfying (FE) and (ES), $\Upsilon_{\varepsilon}(u)\leq\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$ implies $u=u^{\varepsilon}_{0}$.

The key-argument is the following: if the initial law is symmetric, then the law at time $t$ is still symmetric. The proof is similar to the previous one, so it is left to the reader’s attention.

Before making the convergence, we need a last result on the adherence values: the free-energy of a limiting value is less than the limit value of the free-energy.

We put $\gamma_{k}^{+}(x):=u_{t_{k}}^{\varepsilon}(x)\log(u_{t_{k}}^{\varepsilon}(x))\mathbh{1}_{\{u_{t_{k}}^{\varepsilon}(x)<1\}}\mathbh{1}_{\{|x|>R\}}$. By proceeding as in the proof of Lemma 1.3, we have

Consequently, it leads to the lower-bound

for all $R>0$. Consequently, $\Upsilon_{\varepsilon}(u_{\infty}^{\varepsilon})\leq L_{0}$. {pf*}Proof of the theorem Plan: The first step of the proof consists of the application of the Prohorov theorem since the family of measure is tight. We shall prove the uniqueness of the adherence value. We will proceed a reductio ad absurdum. The previous results provide $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}\neq\varnothing$ where $\mathcal{A}$ is introduced in Definition 1.7. We will then study all the possible cases, and we will prove that all of these cases imply contradictions. If $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{+}\}$ and $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{-}\}$ imply contradiction since $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$ are the unique minimizers of the free-energy. The cases $u^{\varepsilon}_{0}\in\mathcal{A}$ and $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}$ contradict (M3). {longlist}[Step 2.1.1.]

As condition (M3) is true, there are exactly three stationary measures: $u^{\varepsilon}_{0}$, $u^{\varepsilon}_{+}$ and $u^{\varepsilon}_{-}$. By Theorem 1.6, we know that $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}\neq\varnothing$. We split this step into three cases:

$\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}$.

$\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{+}\}$.

By symmetry, we will not deal with the case $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{-}\}$.

We will prove that the first case, $u^{\varepsilon}_{0}\in\mathcal{A}$, is impossible. It will be the core of the proof.

We deal now with the third case, $\mathcal{A}\cap\{u^{\varepsilon}_{0};u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}=\{u^{\varepsilon}_{+};u^{\varepsilon}_{-}\}$.

Basins of attraction

Now we shall provide some conditions in order to precise the limit.

Let $du_{0}$ be a symmetric probability measure which verifies (FE) and (ES). We assume that $V^{\prime\prime}(0)+F^{\prime\prime}(0)\neq 0$. Then, for $\varepsilon$ small enough $u_{t}^{\varepsilon}$ converges weakly toward $u^{\varepsilon}_{0}$.

$V^{\prime\prime}(0)+F^{\prime\prime}(0)\neq 0$, and both functions $V^{\prime\prime}$ and $F^{\prime\prime}$ are convex. Theorem 1.6 in HT3 implies the existence and the uniqueness of a symmetric stationary measure $u^{\varepsilon}_{0}$ for $\varepsilon$ small enough.

Theorem 1.6 provides the existence of a stationary measure $u^{\varepsilon}$ and an increasing sequence $(t_{k})_{k}$ which goes to $\infty$ such that $u_{t_{k}}^{\varepsilon}$ converges weakly toward $u^{\varepsilon}$ and $\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})$ converges toward $\Upsilon_{\varepsilon}(u^{\varepsilon})$. As $u_{t}^{\varepsilon}$ is symmetric for all $t\geq 0$, we deduce $u^{\varepsilon}=u^{\varepsilon}_{0}$, the unique symmetric stationary measure.

We proceed a reductio ad absurdum by assuming the existence of another sequence $(s_{k})_{k}$ which goes to $\infty$ such that $u_{s_{k}}^{\varepsilon}$ does not converge toward $u^{\varepsilon}_{0}$. The uniform boundedness of the second moment with respect to the time permits to extract a subsequence [we continue to write $(s_{k})_{k}$ for simplifying] such that $u_{s_{k}}^{\varepsilon}$ converges weakly toward $u_{\infty}^{\varepsilon}\neq u^{\varepsilon}_{0}$. Proposition 2.5 implies $\Upsilon_{\varepsilon}(u_{\infty}^{\varepsilon})\leq\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})$. Lemma 2.4 implies $u_{\infty}^{\varepsilon}=u^{\varepsilon}_{0}$. This is absurd.

We assume $V^{\prime\prime}(0)+F^{\prime\prime}(0)\neq 0$ in order to have a unique symmetric stationary measure for $\varepsilon$ small enough, that is to say (0M1). We can extend to the case $V^{\prime\prime}(0)+F^{\prime\prime}(0)=0$ by using auniform propagation of chaos; see Theorem 6.5 in TT . We can also assume that $n=2$ which means $\deg(F)=4$ by Section 4.2 in HT1 .

In the previous theorem, if we assumed (FM) instead of (ES), we could have directly applied Theorem 2.1.

The principal tool of the previous theorem is the stability of a subset (all the symmetric measures with a finite $8q^{2}$-moment). If we could find an invariant subset which contains $u^{\varepsilon}_{+}$, but neither $u^{\varepsilon}_{0}$ nor $u^{\varepsilon}_{-}$, we could apply the same method than previously.

Instead of this, we will consider an inequality linked to the free-energy and we will exhibit a simple subset included in the domain of attraction of $u^{\varepsilon}_{+}$. Let us first introduce the following hyperplan:

Let $du_{0}$ be a probability measure which verifies (FE) and (FM). We assume also

Under (M3), $u_{t}^{\varepsilon}$ converges weakly toward $u^{\varepsilon}_{+}$.

We know by Theorem 2.1 that there exists a stationary measure $u^{\varepsilon}$ such that $(u_{t}^{\varepsilon})_{t}$ converges weakly toward $u^{\varepsilon}$. And, by Proposition 1.8, $\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})$ converges toward $\Upsilon_{\varepsilon}(u^{\varepsilon})$. {longlist}[Step 1.]

We deduce $u^{\varepsilon}\neq u^{\varepsilon}_{0}$ since $t\mapsto\xi(t)=\Upsilon_{\varepsilon}(u_{t}^{\varepsilon})$ is nonincreasing.

which contradicts the fact that $\xi$ is nonincreasing.

Assumption (M3) implies the weak convergence toward $u^{\varepsilon}_{+}$. ∎ \noqed We use now Theorem 3.4 in some particular cases.

Let $du_{0}$ be a probability measure which verifies (FE) and (FM). We assume also

where $x_{0}$ is defined in the Introduction. Under either conditions (LIN) or (SYN), $u_{t}^{\varepsilon}$ converges weakly toward $u^{\varepsilon}_{+}$ for $\varepsilon$ small enough.

Step 1. Theorem 3.2 in HT1 and Theorem 1.11 imply condition (M3) under (LIN) or (SYN). {longlist}[Step 2.]

Lemma .3 provides the limit $\lim_{\varepsilon\longrightarrow 0}\Upsilon_{\varepsilon}(u^{\varepsilon}_{0})=V(x_{0})+\frac{1}{4}F(2x_{0})$. Then, we deduce

We split now the study depending on whether we use conditions (LIN) or (SYN): {longlist}[(SYN)]

If $F^{\prime}$ is linear, $\frac{F^{\prime\prime}(0)}{2}x^{2}=\frac{1}{4}F(2x)$. So the minimum of $x\mapsto V(x)+\frac{1}{4}F(2x)$ is $V(x_{0})+\frac{1}{4}F(2x_{0})$. We can easily prove that

for all $u\in\mathcal{H}$. Then, $\lim_{\varepsilon\longrightarrow 0}\min_{u\in\mathcal{H}}\Upsilon_{\varepsilon}(u)\geq V(x_{0})+\frac{1}{4}F(2x_{0})$. Inequality (3.1) provides $\lim_{\varepsilon\longrightarrow 0}\inf_{u\in\mathcal{H}}\Upsilon_{\varepsilon}(u)=V(x_{0})+\frac{1}{4}F(2x_{0})$.

Since $V^{\prime\prime}(0)+F^{\prime\prime}(0)>0$, (3.2) implies $\Upsilon_{\varepsilon}(u)\geq-\frac{\varepsilon}{4}-\frac{4\varepsilon}{\exp(1)}$ for all $u\in\mathcal{H}$ if $\varepsilon$ is less than $2(V^{\prime\prime}(0)+F^{\prime\prime}(0))$. We deduce that$\lim_{\varepsilon\longrightarrow 0}\inf_{u\in\mathcal{H}}\Upsilon_{\varepsilon}(u)\geq 0$. However, as $V^{\prime\prime}(0)+F^{\prime\prime}(0)>0$, Theorem 5.4 in HT2 implies $x_{0}=0$ so $V(x_{0})+\frac{1}{4}F(2x_{0})=0$. Inequality (3.1) provides the following limit: $\lim_{\varepsilon\longrightarrow 0}\inf_{u\in\mathcal{H}}\Upsilon_{\varepsilon}(u)=0=V(x_{0})+\frac{1}{4}F(2x_{0})$.

Consequently, $\Upsilon_{\varepsilon}(u_{0})<\inf_{u\in\mathcal{H}}\Upsilon_{\varepsilon}(u)$ for $\varepsilon$ small enough. Then, we apply Theorem 3.4. ∎ \noqed

Appendix: Useful technical results

In this annex, we present some results used previously in the proofs of the main theorems.

Proposition .1 allows us to ensure that even if the free-energy does not reach its global minimum on the stationary measure $u^{\varepsilon}_{0}$, if the unique symmetric stationary measure is an adherence value, then it is unique.

Proposition .2 is a general result on the self-stabilizing processes. Indeed, it is well known that $du_{t}^{\varepsilon}$ is absolutely continuous with respect to the Lebesgue measure for all $t>0$. Proposition .2 extends this instantaneous regularization to the finiteness of all the moments.

Lemma .3 consists in asymptotic computation of the free-energy in the small-noise limit for some useful measures. Lemma .4 use a Laplace method for making a tedious computation which is necessary for avoiding to assume that each family of stationary measures verify condition (H).

We present now the essential proposition for proving Theorem 2.1.

Let $du_{0}$ be a probability measure which verifies (FE) and (FM). We assume the existence of two polynomial functions $\mathcal{P}$ and $\mathcal{Q}$, a smooth function $\varphi$ with compact support such that $|\varphi(x)|\leq\mathcal{P}(x)$ and $|\varphi^{\prime}(x)|^{2}\leq\mathcal{Q}(x)$, $\kappa>0$ and two sequences $(r_{k})_{k}$ and $(s_{k})_{k}$ which go to $\infty$ such that for all $r_{k}\leq t\leq s_{k}<r_{k+1}$,

Step 1. We will prove that $\liminf_{k\longrightarrow+\infty}(s_{k}-r_{k})>0$. We introduce the function

This function is well defined since $|\varphi|$ is bounded by a polynomial function. The derivation of $\Phi$, the use of equation (III) and an integration by parts lead to

By definition of the two sequences $(r_{k})_{k}$ and $(s_{k})_{k}$, we have

Combining this identity with (.1), it yields

We apply the Cauchy–Schwarz inequality, and we obtain

since $\xi$ is nonincreasing; see Proposition 1.2. Moreover, $\xi(t)$ converges as $t$ goes to $\infty$; see Lemma 1.4. It implies the convergence of $\xi(r_{k})-\xi(s_{k})$ toward when $k$ goes to $+\infty$. Consequently, $s_{k}-r_{k}$ converges toward $+\infty$ so $\liminf_{k\longrightarrow+\infty}s_{k}-r_{k}>0$. {longlist}[Step 2.]

Let $du_{0}$ be a probability measure which verifies (FE) and (ES). Then, for all $t>0$, $du_{t}^{\varepsilon}$ satisfies (FM).

Consequently, the application $x\mapsto V^{\prime}(x)+F^{\prime}\ast u_{t}^{\varepsilon}(x)$ is a polynomial function with degree $2q-1$. Furthermore, the principal term does not depend of the moments of the law $du_{t}^{\varepsilon}$, so we can write

where $C_{l}$ is a positive constant. The application of Ito formula provides

after using (.2). We choose $l:=l_{0}+1-q$, and then we take the expectation. We obtain

Let $\varepsilon_{0}$ such that there exist three families of stationary measures $(u^{\varepsilon}_{+})_{\varepsilon\in]0;\varepsilon_{0}]}$, $(u^{\varepsilon}_{-})_{\varepsilon\in]0;\varepsilon_{0}]}$ and $(u^{\varepsilon}_{0})_{\varepsilon\in]0;\varepsilon_{0}]}$ which verify

where $x_{0}$ is defined in the Introduction. Then, we have the following limits:

Plus, by considering the measure $v^{\varepsilon}_{+}(x):=Z^{-1}\exp[-\frac{2}{\varepsilon}(V(x)+F(x-a))]$, we have

Step 1. We begin to prove the result for $u^{\varepsilon}_{0}$. {longlist}[Step 1.1.]

We can write $u^{\varepsilon}_{0}(x)=Z^{-1}\exp[-\frac{2}{\varepsilon}(V(x)+F\ast u^{\varepsilon}_{0}(x))]$ since it is a stationary measure. Hence

If $V^{\prime\prime}(0)+F^{\prime\prime}(0)\neq 0$, we can apply Lemma A.4 in HT3 to $f(x):=1$ and $U_{\varepsilon}(x):=V(x)+F\ast u^{\varepsilon}_{0}(x)$. This provides

where the constant $C_{\varepsilon}$ verifies $\varepsilon\log(C_{\varepsilon})\longrightarrow 0$ in the small-noise limit. We deduce

when $\varepsilon$ collapses. Consequently, it leads to the following limit:

We assume now $V^{\prime\prime}(0)+F^{\prime\prime}(0)=0$. Then $x_{0}=0$ according to Proposition 3.7 and Remark 3.8 in HT2 . Propositions 3.5 and 3.6 in HT3 imply

when $\varepsilon$ collapses. Consequently, we obtain the following limit:

We prove now the result for $u^{\varepsilon}_{+}$ (the proof is similar for $u^{\varepsilon}_{-}$).

We can write $u^{\varepsilon}_{+}(x)=Z^{-1}\exp[-\frac{2}{\varepsilon}(V(x)+F\ast u^{\varepsilon}_{+}(x))]$ since it is a stationary measure. Hence

where the constant $C_{\varepsilon}$ verifies $\varepsilon\log(C_{\varepsilon})\longrightarrow 0$ in the small-noise limit. We deduce

when $\varepsilon\longrightarrow 0$. Consequently, the following limit holds:

We proceed similarly for $v^{\varepsilon}_{+}$. ∎ \noqed We provide here a useful asymptotic result linked to the Laplace method.

$U_{k}$ has exactly one global minimum location $A_{j}^{(k)}$ on each interval $I_{j}$, where $I_{j}$ represents the Voronoï cells corresponding to the central points $A_{j}$, with $1\leq j\leq r$.

$A_{j}^{(k)}$ tends toward $A_{j}$ when $k$ goes to $+\infty$.

(1) The first point of the lemma is exactly the one of Lemma A.4 in HT3 . {longlist}[(2)]

Since $U_{k}(x)\geq x^{2}$ for $|x|\geq R$ and $k>k_{c}$, we can confine each $A_{j}^{(k)}$ in a compact subset. Then, the uniform convergence on all the compact subset implies the convergence of $A_{j}^{(k)}$ toward $A_{j}$ when $k$ goes to $+\infty$.

Let $\rho>0$ arbitrarily small such that $[A_{j}-\rho,A_{j}+\rho]\subset I_{j}$. For obvious reasons, we can extract a subsequence such that

with $\lambda_{i}(\rho)\geq 0$ for all $1\leq i\leq r$ and $\sum_{j=1}^{r}\lambda_{j}(\rho)=1$.

We can note that the generation of the sequence $\psi(k)$ depends on the choice of $\rho$. Consequently, in the following, we can take $\rho$ arbitrarily small, then $\varepsilon_{\psi(k)}$ arbitrarily small.

As the $r$ families $(\lambda_{j}(\rho))_{\rho>0}$ are bounded, we can extract a subsequence $(\rho_{p})_{p}$ such that $\lambda_{j}(\rho_{p})$ tends toward $\lambda_{j}$ when $p$ goes to $+\infty$. Furthermore, $\lambda_{j}\geq 0$ for all $1\leq j\leq r$ and $\sum_{j=1}^{r}\lambda_{j}=1$. For simplifying, we will write $\rho$ (resp., $k$) instead of $\rho_{p}$ [resp., $\psi(k)$].

Let $\tau>0$ arbitrarily small. We take $R\geq 2$ such that

The convergence of $\lambda_{j}(\rho)$ toward $\lambda_{j}$ implies the existence of $\rho_{0}>0$ such that for all $\rho<\rho_{0}$, we have

By taking $\rho<\min\{\rho_{0};\min_{1\leq l\leq N}\frac{\tau}{5lR^{l-1}}\}$, we deduce

We will prove that the third term tends toward . It is sufficient to prove the following convergence:

for all $1\leq j\leq r$. Since $I_{j}\subset[-R,R]$, we have

Let us prove the convergence toward of the right-hand term:

Let $\rho_{1}>0$ such that for all $\rho<\rho_{1}$, we have

We take $\rho<\min\{\rho_{0},\rho_{1},\min_{1\leq l\leq N}\frac{\tau}{5lR^{l-1}}\}$. As $U_{k}$ converges uniformly toward $U$ on all the compact subset, we deduce that for $k\geq k_{0}$, we have

By using the growth property on $U_{k}$ then the change of variable $x:=\sqrt{\varepsilon_{k}}y$, it yields

where $C(l)$ is a constant. We recall the assumption $\max_{z\in[A_{1}-1;A_{1}+1]}U(z)+2<\frac{R^{2}}{2}$. Since $U_{k}$ converges toward $U$ uniformly on each compact subset, we have $\max_{z\in[A_{1}-1;A_{1}+1]}U_{k}(z)+1<\frac{R^{2}}{2}$ for $k\geq k_{1}$ (independently of $\rho$). Consequently,

For $k\geq k_{2}$, we have the inequality

By taking $k\geq\max\{k_{0},k_{1},k_{2}\}$, we obtain

By taking $\rho<\min\{\rho_{0},\rho_{1},\frac{\tau}{5lR^{l-1}}\}$ and $k\geq\max\{k_{0},k_{1},k_{2}\}$, inequalities (.3)–(.6) and (.7) provide

for all $1\leq l\leq N$. This achieves the proof. ∎ \noqed

This lemma seems weaker than Lemma A.4 in HT3 . However, here, we do not assume that the second derivative of $U$ is positive in all the global minimum locations.

Acknowledgment

It is a great pleasure to thank Samuel Herrmann for his remarks concerning this work. Most of the ideas for this work were found while I was at the Institut Élie Cartan in Nancy. And so, I want to mention that I would not have been able to write this paper without the hospitality I received from the beginning.

Finalement, un très grand merci à Manue et à Sandra pour tout.

References